Convergence of the Dirichlet–Neumann alternating method for semilinear elliptic equations
(2025) In SIAM Journal on Numerical Analysis 63(5). p.2133-2154- Abstract
- The Dirichlet–Neumann alternating method is a common domain decomposition method for nonoverlapping domain decompositions without cross-points, and the method has been studied extensively for linear elliptic equations. However, for nonlinear elliptic equations, there are only convergence results for some specific cases in one spatial dimension. The aim of this manuscript is therefore to prove that the Dirichlet–Neumann alternating method converges for a class of semilinear elliptic equations on Lipschitz continuous domains in two and three spatial dimensions. This is achieved by first proving a new result on the convergence of nonlinear iterations in Hilbert spaces and then applying this result to the Steklov–Poincaré formulation of the... (More)
- The Dirichlet–Neumann alternating method is a common domain decomposition method for nonoverlapping domain decompositions without cross-points, and the method has been studied extensively for linear elliptic equations. However, for nonlinear elliptic equations, there are only convergence results for some specific cases in one spatial dimension. The aim of this manuscript is therefore to prove that the Dirichlet–Neumann alternating method converges for a class of semilinear elliptic equations on Lipschitz continuous domains in two and three spatial dimensions. This is achieved by first proving a new result on the convergence of nonlinear iterations in Hilbert spaces and then applying this result to the Steklov–Poincaré formulation of the Dirichlet–Neumann alternating method. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/60a1f2f7-2f73-47eb-982a-568c2de89472
- author
- Engström, Emil LU
- organization
- publishing date
- 2025-10-14
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- nonlinear domain decomposition, Dirichlet-Neumann method, linear convergence, semilinear elliptic equation
- in
- SIAM Journal on Numerical Analysis
- volume
- 63
- issue
- 5
- pages
- 22 pages
- publisher
- Society for Industrial and Applied Mathematics
- external identifiers
-
- scopus:105019490035
- ISSN
- 0036-1429
- DOI
- 10.1137/24M1703550
- project
- Next generation numerical partitioning schemes for time dependent PDEs
- language
- English
- LU publication?
- yes
- id
- 60a1f2f7-2f73-47eb-982a-568c2de89472
- date added to LUP
- 2026-01-14 14:37:46
- date last changed
- 2026-01-15 04:01:35
@article{60a1f2f7-2f73-47eb-982a-568c2de89472,
abstract = {{The Dirichlet–Neumann alternating method is a common domain decomposition method for nonoverlapping domain decompositions without cross-points, and the method has been studied extensively for linear elliptic equations. However, for nonlinear elliptic equations, there are only convergence results for some specific cases in one spatial dimension. The aim of this manuscript is therefore to prove that the Dirichlet–Neumann alternating method converges for a class of semilinear elliptic equations on Lipschitz continuous domains in two and three spatial dimensions. This is achieved by first proving a new result on the convergence of nonlinear iterations in Hilbert spaces and then applying this result to the Steklov–Poincaré formulation of the Dirichlet–Neumann alternating method.}},
author = {{Engström, Emil}},
issn = {{0036-1429}},
keywords = {{nonlinear domain decomposition; Dirichlet-Neumann method; linear convergence; semilinear elliptic equation}},
language = {{eng}},
month = {{10}},
number = {{5}},
pages = {{2133--2154}},
publisher = {{Society for Industrial and Applied Mathematics}},
series = {{SIAM Journal on Numerical Analysis}},
title = {{Convergence of the Dirichlet–Neumann alternating method for semilinear elliptic equations}},
url = {{http://dx.doi.org/10.1137/24M1703550}},
doi = {{10.1137/24M1703550}},
volume = {{63}},
year = {{2025}},
}